When was the first football championship game?

In the NFL, that would have been 1933. The NFL had only one division between 1920-1932 and whichever team had the highest winning percentage at the end of the season was crowned champion. In 1933, the NFL split teams into two divisions and instituted a championship game to determine the league champion. In the first championship game in 1933, the Chicago Bears defeated the New York Giants 23-21.

In the 1932 season, there was a tie for the championship and the NFL scheduled a playoff game to determine who would be the league champion. In that game, the Chicago Bears defeated the Portsmouth Spartans (now the Detroit Lions) 9-0.